PENELOPE 2003 - OECD Nuclear Energy Agency

4.1. Elastic scattering 129
simulated electron track when it reaches the interface is
1 − 〈cos χ〉 = t ( ) 1 − 〈cos χ〉
(s)
≃ t
s
λ (s)
el,1
, (4.23)
which practically coincides with the exact mean deviation after the path length t within
region “1”, as required. Thus, by sampling the position of the soft collision uniformly in
the segment (0, s) we make sure that the electron reaches the interface with the correct
average direction of movement.
Angular deflections in soft scattering events
In the random hinge method, the global effect of the soft collisions experienced by
the particle along a path segment of length s between two consecutive hard events is
simulated as a single artificial soft scattering event. The angular deflection follows the
multiple scattering distribution F (s) (s; χ). Unfortunately, the exact Legendre expansion,
eq. (4.17), is not appropriate for Monte Carlo simulation, since this expansion converges
very slowly (because the associated single scattering DCS is not continuous) and the
sum varies rapidly with the path length s.
Whenever the cutoff angle θ c is small, the distribution F (s) (s; χ) may be calculated
by using the small angle approximation (see e.g. Lewis, 1950). Notice that θ c can be
made as small as desired by selecting a small enough value of C 1 , see eqs. (4.9) and
(4.10). Introducing the limiting form of the Legendre polynomials
into eq. (4.16a) we get
1
λ (s)
el,l
P l (cos θ) ≃ 1 − 1 4 l(l + 1)θ2 (4.24)
∫
l(l + 1) θc
= N 2π θ 2 dσ el(θ)
4 0 dΩ
sin θ dθ =
l(l + 1)
2
1
λ (s)
el,1
, (4.25)
i.e. the transport mean free paths λ (s)
el,l are completely determined by the single value
λ (s)
el,1. The angular distribution F (s) then simplifies to
F (s) (s; χ) =
∞∑
l=0
2l + 1
4π
⎡
exp l(l + 1)
⎣−
2
s
λ (s)
el,1
⎤
⎦ P l (cos χ). (4.26)
This expression can be evaluated by using the Molière (1948) approximation for the
Legendre polynomials, we obtain (see Fernández-Varea et al., 1993b)
⎡
exp ⎣
s
F (s) (s; χ) = 1 ( χ
2π sin χ
) 1/2
λ (s)
el,1
s
8λ (s)
el,1
− λ(s) el,1
2s χ2 ⎤
⎦ , (4.27)
which does not differ significantly from the Gaussian distribution with variance s/λ (s)
el,1.
This result is accurate whenever s ≪ λ (s)
el,1 and θ c ≪ 1. It offers a possible method